Chapter 37 The MIXED Procedure. Chapter Table of Contents

Transcription

1 Chapter 37 The MIXED Procedure Chapter Table of Contents OVERVIEW BasicFeatures NotationfortheMixedModel PROCMIXEDContrastedwithOtherSASProcedures GETTING STARTED ClusteredDataExample SYNTAX PROCMIXEDStatement BYStatement CLASSStatement CONTRAST Statement ESTIMATE Statement IDStatement LSMEANSStatement MAKEStatement MODELStatement PARMSStatement PRIORStatement RANDOM Statement REPEATED Statement WEIGHTStatement DETAILS MixedModelsTheory ParameterizationofMixedModels DefaultOutput OutputChangesinVersion ComputationalIssues EXAMPLES Example 37.1 Split-Plot Design Example 37.2 Repeated Measures Example 37.3 Plotting the Likelihood Example37.4KnownGandR

2 1946 Chapter 37. The MIXED Procedure Example 37.5 Random Coefficients Example 37.6 Line-Source Sprinkler Irrigation REFERENCES

3 Chapter 37 The MIXED Procedure Overview The MIXED procedure fits a variety of mixed linear models to data and enables you to use these fitted models to make statistical inferences about the data. A mixed linear model is a generalization of the standard linear model used in the GLM procedure, the generalization being that the data are permitted to exhibit correlation and nonconstant variability. The mixed linear model, therefore, provides you with the flexibility of modeling not only the means of your data (as in the standard linear model) but their variances and covariances as well. The primary assumptions underlying the analyses performed by PROC MIXED are as follows: The data are normally distributed (Gaussian). The means (expected values) of the data are linear in terms of a certain set of parameters. The variances and covariances of the data are in terms of a different set of parameters, and they exhibit a structure matching one of those available in PROC MIXED. Since Gaussian data can be modeled entirely in terms of their means and variances/covariances, the two sets of parameters in a mixed linear model actually specify the complete probability distribution of the data. The parameters of the mean model are referred to as fixed-effects parameters, and the parameters of the variancecovariance model are referred to as covariance parameters. The fixed-effects parameters are associated with known explanatory variables, as in the standard linear model. These variables can be either qualitative (as in the traditional analysis of variance) or quantitative (as in standard linear regression). However, the covariance parameters are what distinguishes the mixed linear model from the standard linear model. The need for covariance parameters arises quite frequently in applications, the following being the two most typical scenarios: The experimental units on which the data are measured can be grouped into clusters, and the data from a common cluster are correlated. Repeated measurements are taken on the same experimental unit, and these repeated measurements are correlated or exhibit variability that changes.

4 1948 Chapter 37. The MIXED Procedure The first scenario can be generalized to include one set of clusters nested within another. For example, if students are the experimental unit, they can be clustered into classes, which in turn can be clustered into schools. Each level of this hierarchy can introduce an additional source of variability and correlation. The second scenario occurs in longitudinal studies, where repeated measurements are taken over time. Alternatively, the repeated measures could be spatial or multivariate in nature. PROC MIXED provides a variety of covariance structures to handle the previous two scenarios. The most common of these structures arises from the use of random-effects parameters, which are additional unknown random variables assumed to impact the variability of the data. The variances of the random-effects parameters, commonly known as variance components, become the covariance parameters for this particular structure. Traditional mixed linear models contain both fixed- and random-effects parameters, and, in fact, it is the combination of these two types of effects that led to the name mixed model. PROC MIXED fits not only these traditional variance component models but numerous other covariance structures as well. PROC MIXED fits the structure you select to the data using the method of restricted maximum likelihood (REML), also known as residual maximum likelihood. It is here that the Gaussian assumption for the data is exploited. Other estimation methods are also available, including maximum likelihood and MIVQUE0. The details behind these estimation methods are discussed in subsequent sections. Once a model has been fit to your data, you can use it to draw statistical inferences via both the fixed-effects and covariance parameters. PROC MIXED computes several different statistics suitable for generating hypothesis tests and confidence intervals. The validity of these statistics depends upon the mean and variance-covariance model you select, so it is important to choose the model carefully. Some of the output from PROC MIXED helps you assess your model and compare it with others. Basic Features PROC MIXED provides easy accessibility to numerous mixed linear models that are useful in many common statistical analyses. In the style of the GLM procedure, PROC MIXED fits the specified mixed linear model and produces appropriate statistics. Some basic features of PROC MIXED are covariance structures, including variance components, compound symmetry, unstructured, AR(1), Toeplitz, spatial, general linear, and factor analytic GLM-type grammar, using MODEL, RANDOM, and REPEATED statements for model specification and CONTRAST, ESTIMATE, and LSMEANS statements for inferences appropriate standard errors for all specified estimable linear combinations of fixed and random effects, and corresponding t- and F-tests subject and group effects that enable blocking and heterogeneity, respectively

5 Notation for the Mixed Model 1949 REML and ML estimation methods implemented with a Newton-Raphson algorithm capacity to handle unbalanced data ability to create a SAS data set corresponding to any table PROC MIXED uses the Output Delivery System (ODS), a SAS subsystem that provides capabilities for displaying and controlling the output from SAS procedures. ODS enables you to convert any of the output from PROC MIXED into a SAS data set. See the Output Changes in Version 7 section on page Notation for the Mixed Model This section introduces the mathematical notation used throughout this chapter to describe the mixed linear model. You should be familiar with basic matrix algebra (refer to Searle 1982). A more detailed description of the mixed model is contained in the Mixed Models Theory section on page A statistical model is a mathematical description of how data are generated. The standard linear model, as used by the GLM procedure, is one of the most common statistical models: y = X + In this expression, y represents a vector of observed data, is an unknown vector of fixed-effects parameters with known design matrix X, and is an unknown random error vector modeling the statistical noise around X. The focus of the standard linear model is to model the mean of y by using the fixed-effects parameters. The residual errors are assumed to be independent and identically distributed Gaussian random variables with mean 0 and variance 2. The mixed model generalizes the standard linear model as follows: y = X + Z + Here, is an unknown vector of random-effects parameters with known design matrix Z, and is an unknown random error vector whose elements are no longer required to be independent and homogeneous. To further develop this notion of variance modeling, assume that and are Gaussian random variables that are uncorrelated and have expectations 0 and variances G and R, respectively. The variance of y is thus V = ZGZ 0 + R

6 1950 Chapter 37. The MIXED Procedure Note that, when R = 2 I and Z = 0, the mixed model reduces to the standard linear model. You can model the variance of the data, y, by specifying the structure (or form) of Z, G, andr. The model matrix Z is set up in the same fashion as X, the model matrix for the fixed-effects parameters. For G and R, you must select some covariance structure. Possible covariance structures include variance components compound symmetry (common covariance plus diagonal) unstructured (general covariance) autoregressive spatial general linear factor analytic By appropriately defining the model matrices X and Z, as well as the covariance structure matrices G and R, you can perform numerous mixed model analyses. PROC MIXED Contrasted with Other SAS Procedures PROC MIXED is a generalization of the GLM procedure in the sense that PROC GLM fits standard linear models, and PROC MIXED fits the wider class of mixed linear models. Both procedures have similar CLASS, MODEL, CONTRAST, ESTI- MATE, and LSMEANS statements, but their RANDOM and REPEATED statements differ (see the following paragraphs). Both procedures use the nonfull-rank model parameterization, although the sorting of classification levels can differ between the two. PROC MIXED computes only Type I Type III tests of fixed effects, while PROC GLM offers Types I IV. The RANDOM statement in PROC MIXED incorporates random effects constituting the vector in the mixed model. However, in PROC GLM, effects specified in the RANDOM statement are still treated as fixed as far as the model fit is concerned, and they serve only to produce corresponding expected mean squares. These expected mean squares lead to the traditional ANOVA estimates of variance components. PROC MIXED computes REML and ML estimates of variance parameters, which are generally preferred to the ANOVA estimates (Searle 1988; Harville 1988; Searle, Casella, and McCulloch 1992). Optionally, PROC MIXED also computes MIVQUE0 estimates, which are similar to ANOVA estimates. The REPEATED statement in PROC MIXED is used to specify covariance structures for repeated measurements on subjects, while the REPEATED statement in PROC GLM is used to specify various transformations with which to conduct the traditional univariate or multivariate tests. In repeated measures situations, the mixed model approach used in PROC MIXED is more flexible and more widely applicable than either the univariate or multivariate approaches. In particular, the mixed model ap-

7 Clustered Data Example 1951 proach provides a larger class of covariance structures and a better mechanism for handling missing values. PROC MIXED subsumes the VARCOMP procedure. PROC MIXED provides a wide variety of covariance structures, while PROC VARCOMP estimates only simple random effects. PROC MIXED carries out several analyses that are absent in PROC VARCOMP, including the estimation and testing of linear combinations of fixed and random effects. The ARIMA and AUTOREG procedures provide more time series structures than PROC MIXED, although they do not fit variance component models. The CALIS procedure fits general covariance matrices, but it does not allow fixed effects as does PROC MIXED. The LATTICE and NESTED procedures fit special types of mixed linear models that can also be handled in PROC MIXED, although PROC MIXED may run slower because of its more general algorithm. The TSCSREG procedure analyzes time-series cross-sectional data, and it fits some structures not available in PROC MIXED. Getting Started Clustered Data Example Consider the following SAS data set as an introductory example: data heights; input Family Gender$ datalines; 1 F 67 1 F 66 1 F 64 1 M 71 1 M 72 2 F 63 2 F 63 2 F 67 2 M 69 2 M 68 2 M 70 3 F 63 3 M 64 4 F 67 4 F 66 4 M 67 4 M 67 4 M 69 run; The response variable Height measures the heights (in inches) of 18 individuals. The individuals are classified according to Family and Gender. You can perform a traditional two-way analysis of variance of these data with the following PROC MIXED code: proc mixed; class Family Gender; model Height = Gender Family Family*Gender; run; The PROC MIXED statement invokes the procedure. The CLASS statement instructs PROC MIXED to consider both Family and Gender as classification variables. Dummy (indicator) variables are, as a result, created corresponding to all of the distinct levels of Family and Gender. For these data, Family has four levels and Gender has two levels.

8 1952 Chapter 37. The MIXED Procedure The MODEL statement first specifies the response (dependent) variable Height. The explanatory (independent) variables are then listed after the equal (=) sign. Here, the two explanatory variables are Gender and Family, and they comprise the main effects of the design. The third explanatory term, Family*Gender, models an interaction between the two main effects. PROC MIXED uses the dummy variables associated with Gender, Family, andfamily*gender to construct the X matrix for the linear model. A column of 1s is also included as the first column of X to model a global intercept. There are no Z or G matrices for this model, and R is assumed to equal 2 I,whereI is an 1818 identity matrix. The RUN statement completes the specification. The coding is precisely the same as with the GLM procedure. However, much of the output from PROC MIXED is different from that produced by PROC GLM. The following is the output from PROC MIXED. Model Information Data Set Dependent Variable Covariance Structure Estimation Method Residual Variance Method Fixed Effects SE Method Degrees of Freedom Method WORK.HEIGHTS Height Diagonal REML Profile Model-Based Residual Figure Model Information The Model Information table describes the model, some of the variables that it involves, and the method used in fitting it. This table also lists the method (profile, factor, or fit) for handling the residual variance. Class Level Information Class Levels Values Family Gender 2 F M Figure Class Level Information The Class Level Information table lists the levels of all variables specified in the CLASS statement. You can check this table to make sure that the data are correct.

9 Clustered Data Example 1953 Dimensions Covariance Parameters 1 Columns in X 15 Columns in Z 0 Subjects 1 Max Obs Per Subject 18 Observations Used 18 Observations Not Used 0 Total Observations 18 Figure Dimensions The Dimensions table lists the sizes of relevant matrices. This table can be useful in determining CPU time and memory requirements. Covariance Parameter Estimates Cov Parm Estimate Residual Figure Covariance Parameter Estimates The Covariance Parameter Estimates table displays the estimate of 2 model. for the Fitting Information Res Log Likelihood Akaike s Information Criterion Schwarz s Bayesian Criterion Res Log Likelihood 41.6 Figure Model Fitting Information The Fitting Information table lists several pieces of information about the fitted mixed model, including values derived from the computed value of the restricted/residual likelihood. Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F Gender Family Family*Gender Figure Tests of Fixed Effects

10 1954 Chapter 37. The MIXED Procedure The Type 3 Tests of Fixed Effects table displays significance tests for the three effects listed in the MODEL statement. The Type III F -statistics and p-values are the same as those produced by the GLM procedure. However, because PROC MIXED uses a likelihood-based estimation scheme, it does not directly compute or display sums of squares for this analysis. The Type 3 test for Family*Gender effect is not significant at the 5% level, but the tests for both main effects are significant. The important assumptions behind this analysis are that the data are normally distributed and that they are independent with constant variance. For these data, the normality assumption is probably realistic since the data are observed heights. However, since the data occur in clusters (families), it is very likely that observations from the same family are statistically correlated, that is, not independent. The methods implemented in PROC MIXED are still based on the assumption of normally distributed data, but you can drop the assumption of independence by modeling statistical correlation in a variety of ways. You can also model variances that are heterogeneous, that is, nonconstant. For the height data, one of the simplest ways of modeling correlation is through the use of random effects. Here the family effect is assumed to be normally distributed with zero mean and some unknown variance. This is in contrast to the previous model in which the family effects are just constants, or fixed effects. Declaring Family as a random effect sets up a common correlation among all observations having the same level of Family. Declaring Family*Gender as a random effect models an additional correlation between all observations that have the same level of both Family and Gender. One interpretation of this effect is that a female in a certain family exhibits more correlation with the other females in that family than with the other males, and likewise for a male. With the height data, this model seems reasonable. The code to fit this correlation model in PROC MIXED is as follows: proc mixed; class Family Gender; model Height = Gender; random Family Family*Gender; run; Note that Family and Family*Gender are now listed in the RANDOM statement. The dummy variables associated with them are used to construct the Z matrix in the mixed model. The X matrix now consists of a column of 1s and the dummy variables for Gender. The G matrix for this model is diagonal, and it contains the variance components for both Family and Family*Gender. TheR matrix is still assumed to equal 2 I,where I is an identity matrix. The output from this analysis is as follows.

11 Clustered Data Example 1955 Model Information Data Set Dependent Variable Covariance Structure Estimation Method Residual Variance Method Fixed Effects SE Method Degrees of Freedom Method WORK.HEIGHTS Height Variance Components REML Profile Model-Based Containment Figure Model Information The Model Information table shows that the containment method is used to compute the degrees of freedom for this analysis. This is the default method when a RANDOM statement is used; see the description of the DDFM= option on page 1979 for more information. Class Level Information Class Levels Values Family Gender 2 F M Figure Class Levels Information The Class Levels Information table is the same as before. Dimensions Covariance Parameters 3 Columns in X 3 Columns in Z 12 Subjects 1 Max Obs Per Subject 18 Observations Used 18 Observations Not Used 0 Total Observations 18 Figure Dimensions The Dimensions table displays the new sizes of the X and Z matrices.

12 1956 Chapter 37. The MIXED Procedure Iteration History Iteration Evaluations -2 Res Log Like Criterion Convergence criteria met. Figure REML Estimation Iteration History The Iteration History table displays the results of the numerical optimization of the restricted/residual likelihood. Six iterations are required to achieve the default convergence criterion of 1E,8. Covariance Parameter Estimates Cov Parm Estimate Family Family*Gender Residual Figure Covariance Parameter Estimates (REML) The Covariance Parameter Estimates table displays the results of the REML fit. The Estimate column contains the estimates of the variance components for Family and Family*Gender, as well as the estimate of 2. Fitting Information Res Log Likelihood Akaike s Information Criterion Schwarz s Bayesian Criterion Res Log Likelihood 71.0 Figure Fitting Information The Fitting Information table contains basic information about the REML fit.

13 Clustered Data Example 1957 Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F Gender Figure Type 3 Tests of Fixed Effects The Type 3 Tests of Fixed Effects table contains a significance test for the lone fixed effect, Gender. Note that the associated p-value is not nearly as significant as in the previous analysis. This illustrates the importance of correctly modeling correlation in your data. An additional benefit of the random effects analysis is that it enables you to make inferences about gender that apply to an entire population of families, whereas the inferences about gender from the analysis where Family and Family*Gender are fixed effects apply only to the particular families in the data set. PROC MIXED thus offers you the ability to model correlation directly and to make inferences about fixed effects that apply to entire populations of random effects.

14 1958 Chapter 37. The MIXED Procedure Syntax The following statements are available in PROC MIXED. PROC MIXED < options > ; BY variables ; CLASS variables ; ID variables ; MODEL dependent = < fixed-effects ></ options > ; RANDOM random-effects < / options > ; REPEATED < repeated-effect >< / options > ; PARMS (value-list) ::: </ options > ; PRIOR < distribution >< / options > ; CONTRAST label < fixed-effect values ::: > < j random-effect values ::: >, ::: </ options > ; ESTIMATE label < fixed-effect values ::: > < j random-effect values ::: ></ options > ; LSMEANS fixed-effects < / options > ; MAKE table OUT=SAS-data-set ; WEIGHT variable ; Itemswithinanglebrackets(<>)areoptional. TheCONTRAST, ESTIMATE, LSMEANS, MAKE, and RANDOM statements can appear multiple times; all other statements can appear only once. The PROC MIXED and MODEL statements are required, and the MODEL statement must appear after the CLASS statement if a CLASS statement is included. The CON- TRAST, ESTIMATE, LSMEANS, RANDOM, and REPEATED statements must follow the MODEL statement. The CONTRAST and ESTIMATE statements must also follow any RANDOM statements. Table 37.1 summarizes the basic functions and important options of each PROC MIXED statement. The syntax of each statement in Table 37.1 is described in the following sections in alphabetical order after the description of the PROC MIXED statement.

15 PROC MIXED Statement 1959 Table Summary of PROC MIXED Statements MODEL RANDOM Statement Description Important Options PROC MIXED invokes the procedure DATA= specifies input data set, METHOD= specifies estimation method BY performs multiple none PROC MIXED analyses in one invocation CLASS declares qualitative variables none that create indica- tor variables in design matrices ID lists additional variables to be included in pre- none dicted values tables specifies dependent variable and fixed effects, setting up X specifies random effects, setting up Z and G S requests solution for fixed-effects parameters, DDFM= specifies denominator degrees of freedom method, OUTP= outputs predicted values to a data set SUBJECT= creates block-diagonality, TYPE= specifies covariance structure, S requests solution for random-effects parameters, G displays estimated G REPEATED sets up R SUBJECT= creates block-diagonality, TYPE= specifies covariance structure, R displays estimated blocks of R, GROUP= enables betweensubject heterogeneity, LOCAL adds a diagonal PARMS specifies a grid of initial values for the covariance parameters PRIOR performs a samplingbased Bayesian analysis for variance component models CONTRAST constructs custom hypothesis tests ESTIMATE constructs custom scalar estimates LSMEANS computes least squares means for classification fixed effects MAKE WEIGHT converts any displayed table into a SAS data set specifies a variable by which to weight R matrix to R HOLD= and NOITER hold the covariance parameters or their ratios constant, PDATA= reads the initial values from a SAS data set NSAMPLE= specifies the sample size, SEED= specifies the starting seed E displays the L matrix coefficients CL produces confidence limits DIFF computes differences of the least squares means, ADJUST= performs multiple comparisons adjustments, AT changes covariates, OM changes weighting, CL produces confidence limits, SLICE= tests simple effects none. Has been superceded by the Output Delivery System (ODS) none

16 1960 Chapter 37. The MIXED Procedure PROC MIXED Statement PROC MIXED < options >; The PROC MIXED statement invokes the procedure. You can specify the following options. ABSOLUTE makes the convergence criterion absolute. By default, it is relative (divided by the current objective function value). See the CONVF, CONVG, and CONVH options in this section for a description of various convergence criteria. ALPHA=number requests that confidence limits be constructed for the covariance parameter estimates with confidence level 1, number. Thevalueofnumber must be between 0 and 1; the default is ASYCORR produces the asymptotic correlation matrix of the covariance parameter estimates. It is computed from the corresponding asymptotic covariance matrix (see the description of the ASYCOV option, which follows). For ODS purposes, the label of the Asymptotic Correlation table is AsyCorr. ASYCOV requests that the asymptotic covariance matrix of the covariance parameters be displayed. By default, this matrix is the observed inverse Fisher information matrix, which equals 2H,1,whereH is the Hessian (second derivative) matrix of the objective function. See the Covariance Parameter Estimates section on page 2025 for more information about this matrix. When you use the SCORING= option and PROC MIXED converges without stopping the scoring algorithm, PROC MIXED uses the expected Hessian matrix to compute the covariance matrix instead of the observed Hessian. For ODS purposes, the label of the Asymptotic Covariance table is AsyCov. CL<=WALD> requests confidence limits for the covariance parameter estimates. A Satterthwaite approximation is used to construct limits for all parameters that have a default lower boundary constraint of zero. These limits take the form b 2 2 ;1,=2 2 b2 2 ;=2 where =2Z 2, Z is the Wald statistic b 2 =se(b 2 ), and the denominators are quantiles of the 2 -distribution with degrees of freedom. Refer to Milliken and Johnson (1992) and Burdick and Graybill (1992) for similar techniques. For all other parameters, Wald Z-scores and normal quantiles are used to construct the limits. The optional =WALD specification requests Wald limits for all parameters.

17 PROC MIXED Statement 1961 The confidence limits are displayed as extra columns in the Covariance Parameter Estimates table. The confidence level is 1, = 0:95 by default; this can be changed with the ALPHA= option. CONVF<=number> requests the relative function convergence criterion with tolerance number. The relative function convergence criterion is jf k, f k,1 j jf k j number where f k is the value of the objective function at iteration k. To prevent the division by jf k j, use the ABSOLUTE option. The default convergence criterion is CONVH, and the default tolerance is 1E,8. CONVG <=number> requests the relative gradient convergence criterion with tolerance number. The relative gradient convergence criterion is max j jg jk j jf k j number where f k is the value of the objective function, and g jk is the jth element of the gradient (first derivative) of the objective function, both at iteration k. To prevent division by jf k j, use the ABSOLUTE option. The default convergence criterion is CONVH, and the default tolerance is 1E,8. CONVH<=number> requests the relative Hessian convergence criterion with tolerance number. The relative Hessian convergence criterion is g k0 H,1g k k jf k j number where f k is the value of the objective function, g k is the gradient (first derivative) of the objective function, and H k is the Hessian (second derivative) of the objective function, all at iteration k. If H k is singular, then PROC MIXED uses the following relative criterion: g 0 kg k jf k j number To prevent the division by jf k j, use the ABSOLUTE option. The default convergence criterion is CONVH, and the default tolerance is 1E,8. COVTEST produces asymptotic standard errors and Wald Z-tests for the covariance parameter estimates.

18 1962 Chapter 37. The MIXED Procedure DATA=SAS-data-set names the SAS data set to be used by PROC MIXED. The default is the most recently created data set. DFBW has the same effect as the DDFM=BW option in the MODEL statement. EMPIRICAL computes the estimated variance-covariance matrix of the fixed-effects parameters by using the asymptotically consistent estimator described in Huber (1967), White (1980), Liang and Zeger (1986), and Diggle, Liang, and Zeger (1994). This estimator is commonly referred to as the sandwich estimator, and it is computed as follows: (X 0 b V,1 X), SX i=1 X 0 i c V i,1 b i b i 0c Vi,1 X i! (X 0 b V,1 X), Here, b i = y i, X i b, S is the number of subjects, and matrices with an i subscript are those for the ith subject. You must include the SUBJECT= option in either a RANDOM or REPEATED statement for this option to take effect. When you specify the EMPIRICAL option, PROC MIXED adjusts all standard errors and test statistics involving the fixed-effects parameters. This changes output in the following tables (listed in Table 37.7 on page 2028): Contrast, CorrB, CovB, Diffs, Estimates, InvCovB, LSMeans, MMEq, MMEqSol, Slices, SolutionF, Tests, Tests1 Tests3. The OUTP= and OUTPM= data sets are also affected. IC displays a table of various information criteria. Four different criteria are computed in four different ways, producing 16 values in all. Table 37.2 displays the four criteria in both larger-is-better and smaller-is-better forms. Table Information Criteria Criteria Larger-is-better Smaller-is-better Reference AIC `, d,2` +2d Akaike (1974) HQIC `, d log log n,2` + 2d log log n Hannan and Quinn (1979) BIC `, d=2 log n,2` + d log n Schwarz (1978) CAIC `, d(log n +1)=2,2` + d(log n +1) Bozdogan (1987) Here ` denotes the maximum value of the (possibly restricted) log likelihood, d the dimension of the model, and n the number of effective observations. In Version 6 of SAS/STAT software, n equals the number of valid observations for maximum likelihood estimation and n, p for restricted maximum likelihood estimation, where p equals the rank of X. In Version 7, n equals the number of effective subjects as displayed in the Dimensions table, unless this value equals 1, in which case n reverts to the Version 6 values. PROC MIXED evaluates the criteria for both forms using d equal to both q and q + p, where q is the effective number of estimated covariance parameters. The value of d has changed in Version 7 in certain instances. In Version 6, when a parameter estimate lies on a boundary constraint, then it is still included in the calculation of

19 PROC MIXED Statement 1963 d, but in Version 7 it is not. The most common example of this behavior is when a variance component is estimated to equal zero. For ODS purposes, the name of the Information Criteria table is InfoCrit. INFO is a default option in Version 7. The creation of the Model Information and Dimensions tables can be suppressed using the NOINFO option. In Version 6, this option displays the Model Information and Dimensions tables. ITDETAILS displays the parameter values at each iteration and enables the writing of notes to the SAS log pertaining to infinite likelihood and singularities during Newton- Raphson iterations. LOGNOTE writes periodic notes to the log describing the current status of computations. It is designed for use with analyses requiring extensive CPU resources. MAXFUNC=number specifies the maximum number of likelihood evaluations in the optimization process. The default is 150. MAXITER=number specifies the maximum number of iterations. The default is 50. METHOD=REML METHOD=ML METHOD=MIVQUE0 METHOD=TYPE1 METHOD=TYPE2 METHOD=TYPE3 specifies the estimation method for the covariance parameters. The REML specification performs residual (restricted) maximum likelihood, and it is the default method. The ML specification performs maximum likelihood, and the MIVQUE0 specification performs minimum variance quadratic unbiased estimation of the covariance parameters. The METHOD=TYPEn specifications apply only to variance component models with no SUBJECT= effects and no REPEATED statement. An analysis of variance table is included in the output, and the expected mean squares are used to estimate the variance components (refer to Chapter 28, The GLM Procedure, for further explanation). The resulting method-of-moment variance component estimates are used in subsequent calculations, including standard errors computed from ESTIMATE and LSMEANS statements. For ODS purposes, the new table names are Type1, Type2, and Type3, respectively.

20 1964 Chapter 37. The MIXED Procedure MMEQ requests that coefficients of the mixed model equations be displayed. These are " X 0 b R,1X X 0 b R,1 Z b 0,1 R X Z b 0,1 R Z,1 Z + G b # ; " X 0 b R,1 y Z 0 b R,1y # assuming that b G is nonsingular. If b G is singular, PROC MIXED produces the following coefficients " # X 0 R b,1x X b 0,1 R Zb G bgz b 0,1 R,1X GZ b 0 R b Zb G + G b ; " X 0 b R,1 y bgz 0 b R,1y See the Estimating and in the Mixed Model section on page 2015 for further information on these equations. MMEQSOL requests that a solution to the mixed model equations be produced, as well as the inverted coefficients matrix. Formulas for these equations are provided in the preceding description of the MMEQ option. When b G is singular, b and a generalized inverse of the left-hand-side coefficient matrix are transformed using b G to produce b and b C, respectively, where b C is a generalized inverse of the left-hand-side coefficient matrix of the original equations. NAMELEN<=number> specifies the length to which long effect names are shortened. The default and minimum value is 20. NOBOUND has the same effect as the NOBOUND option in the PARMS statement (see page 1985). NOCLPRINT<=number> suppresses the display of the Class Level Information table if you do not specify number. If you do specify number, only levels with totals that are less than number are listed in the table. NOINFO suppresses the display of the Model Information and Dimensions tables. NOITPRINT suppresses the display of the Iteration History table. NOPROFILE includes the residual variance as part of the Newton-Raphson iterations. By default, the residual variance is profiled out of the likelihood. This option may be useful in conjunction with the HOLD= or NOITER option in the PARMS statement. #

21 PROC MIXED Statement 1965 ORD displays ordinates of the relevant distribution in addition to p-values. The ordinate can be viewed as an approximate odds ratio of hypothesis probabilities. ORDER=DATA ORDER=FORMATTED ORDER=FREQ ORDER=INTERNAL specifies the sorting order for the levels of the classification variables (specified in the CLASS statement). This ordering determines which parameters in the model correspond to each level in the data, so the ORDER= option may be useful when you use a CONTRAST or an ESTIMATE statement. The following table shows how PROC MIXED interprets values of the ORDER= option. Value of ORDER= DATA FORMATTED FREQ INTERNAL Levels Sorted by order of appearance in the input data set external formatted value descending frequency count; levels with the most observations come first in the order internal machine representation By default, ORDER=FORMATTED. For FORMATTED and INTERNAL, the sort order is machine dependent. For FORMATTED, the option applies to all classification variables, not just the ones for which you have explicitly defined formats. RATIO produces the ratio of the covariance parameter estimates to the estimate of the residual variance when the latter exists in the model. RIDGE=number specifies the starting value for the minimum ridge value used in the Newton-Raphson algorithm. The default is SCORING<=number> requests that Fisher scoring be used in association with the estimation method up to iteration number, which is 0 by default. When you use the SCORING= option and PROC MIXED converges without stopping the scoring algorithm, PROC MIXED uses the expected Hessian matrix to compute approximate standard errors for the covariance parameters instead of the observed Hessian. The output from the ASYCOV and ASYCORR options is similarly adjusted. SIGITER is an alias for the NOPROFILE option. UPDATE is an alias for the LOGNOTE option.

22 1966 Chapter 37. The MIXED Procedure BY Statement BY variables ; You can specify a BY statement with PROC MIXED to obtain separate analyses on observations in groups defined by the BY variables. When a BY statement appears, the procedure expects the input data set to be sorted in order of the BY variables. The variables are one or more variables in the input data set. If your input data set is not sorted in ascending order, use one of the following alternatives: Sort the data using the SORT procedure with a similar BY statement. Specify the BY statement options NOTSORTED or DESCENDING in the BY statement for the MIXED procedure. The NOTSORTED option does not mean that the data are unsorted but rather means that the data are arranged in groups (according to values of the BY variables) and that these groups are not necessarily in alphabetical or increasing numeric order. Create an index on the BY variables using the DATASETS procedure (in base SAS software). Since sorting the data changes the order in which PROC MIXED reads observations, the sorting order for the levels of the CLASS variable may be affected if you have specified ORDER=DATA in the PROC MIXED statement. This, in turn, affects specifications in the CONTRAST statements. For more information on the BY statement, refer to the discussion in SAS Language Reference: Concepts. For more information on the DATASETS procedure, refer to the discussion in the SAS Procedures Guide. CLASS Statement CLASS variables ; The CLASS statement names the classification variables to be used in the analysis. If the CLASS statement is used, it must appear before the MODEL statement. Classification variables can be either character or numeric. The procedure uses only the first 16 characters of a character variable. Class levels are determined from the formatted values of the CLASS variables. Thus, you can use formats to group values into levels. Refer to the discussion of the FORMAT procedure in the SAS Procedures Guide and to the discussions of the FORMAT statement and SAS formats in SAS Language Reference: Dictionary. You can adjust the display order of CLASS variable levels with the ORDER= option in the PROC MIXED statement.

23 CONTRAST Statement 1967 CONTRAST Statement CONTRAST label < fixed-effect values... > < j random-effect values... >,...< / options > ; The CONTRAST statement provides a mechanism for obtaining custom hypothesis tests. It is patterned after the CONTRAST statement in PROC GLM, although it has been extended to include random effects. This enables you to select an appropriate inference space (McLean, Sanders, and Stroup 1991). You can test the hypothesis L 0 = 0, wherel 0 = (K 0 M 0 ) and 0 = ( 0 0 ), in several inference spaces. The inference space corresponds to the choice of M. When M = 0, your inferences apply to the entire population from which the random effects are sampled; this is known as the broad inference space. When all elements of M are nonzero, your inferences apply only to the observed levels of the random effects. This is known as the narrow inference space, and you can also choose it by specifying all of the random effects as fixed. The GLM procedure uses the narrow inference space. Finally, by zeroing portions of M corresponding to selected main effects and interactions, you can choose intermediate inference spaces. The broad inference space is usually the most appropriate, and it is used when you do not specify any random effects in the CONTRAST statement. In the CONTRAST statement, label identifies the contrast in the table. A label is required for every contrast specified. Labels can be up to 20 characters and must be enclosed in single quotes. fixed-effect identifies an effect that appears in the MODEL statement. The keyword INTERCEPT can be used as an effect when an intercept is fitted in the model. You do not need to include all effects that are in the MODEL statement. random-effect values identifies an effect that appears in the RANDOM statement. The first random effect must follow a vertical bar (j); however, random effects do not have to be specified. are constants that are elements of the L matrix associated with the fixed and random effects. The rows of L 0 are specified in order and are separated by commas. The rows of the K 0 component of L 0 are specified on the left side of the vertical bars (j). These rows test the fixed effects and are, therefore, checked for estimability. The rows of the M 0 component of L 0 are specified on the right side of the vertical bars. They test the random effects, and no estimability checking is necessary. If PROC MIXED finds the fixed-effects portion of the specified contrast to be nonestimable (see the SINGULAR= option on page 1969), then it displays Non-est for the contrast entries.

24 1968 Chapter 37. The MIXED Procedure The following CONTRAST statement reproduces the F-test for the effect A in the split-plot example (see Example 37.1 on page 2037): contrast A broad A A*B , A A*B / df=6; Note that no random effects are specified in the preceding contrast; thus, the inference space is broad. The resulting F-test has two numerator degrees of freedom because L 0 has two rows. The denominator degrees of freedom is, by default, the residual degrees of freedom (9), but the DF= option changes the denominator degrees of freedom to 6. The following CONTRAST statement reproduces the F-test for A when Block and A*Block are considered fixed effects (the narrow inference space): contrast A narrow A A*B A*Block , A A*B A*Block ; The preceding contrast does not contain coefficients for B and Block because they cancel out in estimated differences between levels of A. Coefficients for B and Block are necessary when estimating the mean of one of the levels of A in the narrow inference space (see Example 37.1 on page 2037). If the elements of L are not specified for an effect that contains a specified effect, then the elements of the specified effect are automatically filled in over the levels of the higher-order effect. This feature is designed to preserve estimability for cases when there are complex higher-order effects. The coefficients for the higher-order effect are determined by equitably distributing the coefficients of the lower-level effect as in the construction of least squares means. In addition, if the intercept is specified, it is distributed over all classification effects that are not contained by any other specified effect. If an effect is not specified and does not contain any specified effects, then all of its coefficients in L are set to 0. You can override this behavior by specifying coefficients for the higher-order effect. If too many values are specified for an effect, the extra ones are ignored; if too few are specified, the remaining ones are set to 0. If no random effects are specified, the vertical bar can be omitted; otherwise, it must be present. If a SUBJECT effect is used in the RANDOM statement, then the coefficients specified for the effects in the RANDOM statement are equitably distributed across the levels of the SUBJECT effect. You can use the E option to see exactly what L matrix is used.

25 CONTRAST Statement 1969 The SUBJECT and GROUP options in the CONTRAST statement are useful for the case when a SUBJECT= or GROUP= variable appears in the RANDOM statement, and you want to contrast different subjects or groups. By default, CONTRAST statement coefficients on random effects are distributed equally across subjects and groups. PROC MIXED handles missing level combinations of classification variables similarly to the way PROC GLM does. Both procedures delete fixed-effects parameters corresponding to missing levels in order to preserve estimability. However, PROC MIXED does not delete missing level combinations for random-effects parameters because linear combinations of the random-effects parameters are always estimable. These conventions can affect the way you specify your CONTRAST coefficients. The CONTRAST statement computes the statistic bb 0 L 0 (L 0 b CL),1 L bb F = rank(l) and approximates its distribution with an F-distribution. In this expression, C b is an estimate of the generalized inverse of the coefficient matrix in the mixed model equations. See the Inference and Test Statistics section on page 2017 for more information on this F-statistic. The numerator degrees of freedom in the F-approximation is rank(l), and the denominator degrees of freedom is taken from the Tests of Fixed Effects table and corresponds to the final effect you list in the CONTRAST statement. You can change the denominator degrees of freedom by using the DF= option. You can specify the following options in the CONTRAST statement after a slash (/). CHISQ requests that the 2 -test be performed in addition to the F-test. DF=number specifies the denominator degrees of freedom for the F-test. The default is the denominator degrees of freedom taken from the Tests of Fixed Effects table and corresponds to the final effect you list in the CONTRAST statement. E requests that the L matrix coefficients for the contrast be displayed. For ODS purposes, the label of this L Matrix Coefficients table is Coefficients. GROUP coeffs GRP coeffs sets up random-effect contrasts between different groups when a GROUP= variable appears in the RANDOM statement. By default, CONTRAST statement coefficients on random effects are distributed equally across groups. SINGULAR=number tunes the estimability checking. If v is a vector, define ABS(v) to be the absolute value of the element of v with the largest absolute value. If ABS(K 0,K 0 T) is greater

26 1970 Chapter 37. The MIXED Procedure than C*number for any row of K 0 in the contrast, then K is declared nonestimable. Here T is the Hermite form matrix (X 0 X), X 0 X, and C is ABS(K 0 ) except when it equals 0, and then C is 1. The value for number must be between 0 and 1; the default is 1E,4. SUBJECT coeffs SUB coeffs sets up random-effect contrasts between different subjects when a SUBJECT= variable appears on the RANDOM statement. By default, CONTRAST statement coefficients on random effects are distributed equally across subjects. ESTIMATE Statement ESTIMATE label < fixed-effect values... > < j random-effect values... >,...< / options > ; The ESTIMATE statement is exactly like a CONTRAST statement, except only onerow L matrices are permitted. The actual estimate, L 0 bp, is displayed along with its approximate standard error. An approximate t-test that L 0 bp = 0 is also produced. PROC MIXED selects the degrees of freedom to match those displayed in the Tests of Fixed Effects table for the final effect you list in the ESTIMATE statement. You can modify the degrees of freedom using the DF= option. If PROC MIXED finds the fixed-effects portion of the specified estimate to be nonestimable, then it displays Non-est for the estimate entries. The following examples of ESTIMATE statements compute the mean of the first level of A in the split-plot example (see Example 37.1 on page 2037) for various inference spaces: estimate A1 mean narrow intercept 1 A 1 B.5.5 A*B.5.5 block A*Block ; estimate A1 mean intermed intercept 1 A 1 B.5.5 A*B.5.5 Block ; estimate A1 mean broad intercept 1 A 1 B.5.5 A*B.5.5; The construction of the L vector for an ESTIMATE statement follows the same rules as listed under the CONTRAST statement.

27 ESTIMATE Statement 1971 You can specify the following options in the ESTIMATE statement after a slash (/). ALPHA=number requests that a t-type confidence interval be constructed with confidence level 1, number. Thevalueofnumber must be between 0 and 1; the default is CL requests that t-type confidence limits be constructed. The confidence level is 0.95 by default; this can be changed with the ALPHA= option. DF=number specifies the degrees of freedom for the t-test and confidence limits. The default is the denominator degrees of freedom taken from the Tests of Fixed Effects table and corresponds to the final effect you list in the ESTIMATE statement. DIVISOR=number specifies a value by which to divide all coefficients so that fractional coefficients can be entered as integer numerators. E requests that the L matrix coefficients be displayed. For ODS purposes, the label of this L Matrix Coefficients table is Coefficients. GROUP coeffs GRP coeffs sets up random-effect contrasts between different groups when a GROUP= variable appears in the RANDOM statement. By default, ESTIMATE statement coefficients on random effects are distributed equally across groups. LOWER LOWERTAILED requests that the p-value for the t-test be based only on values less than the t-statistic. A two-tailed test is the default. A lower-tailed confidence limit is also produced if you specify the CL option. SINGULAR=number tunes the estimability checking as documented for the CONTRAST statement. SUBJECT coeffs SUB coeffs sets up random-effect contrasts between different subjects when a SUBJECT= variable appears in the RANDOM statement. By default, ESTIMATE statement coefficients on random effects are distributed equally across subjects. For example, the ESTIMATE statement in the following code from Example 37.5 constructs the difference between the random slopes of the first two batches. proc mixed data=rc; class batch; model y = month / s; random int month / type=un sub=batch s; estimate slope b1 - slope b2 month 1 / subject 1-1; run;